The goal of this work is to determine the implications for DM of a freshly proposed dynamical –and technically natural– scenario [1]}, [2]}, which solves the strong CP problem with an axion much lighter than the canonical QCD one.
Its key point was to assume that Nature is endowed with a \(Z_{\cal N}\)  symmetry realized non-linearly by the axion field [1]}, [2]}. \({\cal N}\)  mirror and degenerate worlds linked by the axion field would coexist with the same coupling strengths as in the SM, with the exception of the
effective \(\theta _k\) -parameters,
\(\mathcal {L}=\sum _{k=0}^{{\cal N}-1} \left[ \mathcal {L}_{\text{SM}_k} +\frac{\alpha _s}{8\pi } \left( \theta _a + \frac{2 \pi k}{{\cal N}} \right) G_k \widetilde{G}_k \right] \,+ \dots \,.\) 
